Definition:Matroid Induced by Linear Independence

Definition

Vector Space

Let $V$ be a vector space.

Let $S$ be a finite subset of $V$.

Let $\mathscr I$ be the set of linearly independent subsets of $S$.

Then the ordered pair $\struct{S, \mathscr I}$ is called a matroid induced on $S$ by linear independence in $V$.

Abelian Group

Let $\struct{G, +}$ be a torsion-free Abelian group.

Let $\struct{G, +, \times}$ be the $\Z$-module associated with $G$.

Let $S$ be a finite subset of $G$.

Let $\mathscr I$ be the set of linearly independent subsets of $S$.

Then the ordered pair $\struct{S, \mathscr I}$ is called the matroid induced by linear independence in $G$ on $S$.